Connect with experts and get insightful answers to your questions on IDNLearn.com. Our experts are available to provide accurate, comprehensive answers to help you make informed decisions about any topic or issue you encounter.
Step-by-step explanation:
If you know, the sine. You must find the cos ratio, using the Pythagorean trig theorem.
Next, you use the half angle identity for sin
[tex] \sin( \frac{ \alpha }{2} ) = \sqrt{ \frac{1 - \cos(a) }{2} } [/tex]
Example.
[tex] \sin( \alpha ) = \frac{ \sqrt{3} }{2} [/tex]
We must find
[tex] \sin( \frac{ \alpha }{2} ) [/tex]
First, use the Pythagorean identity
[tex] \sin {}^{2} ( \alpha ) + \cos {}^{2} ( \alpha ) = 1[/tex]
[tex]( \frac{ \sqrt{3} }{2} ) {}^{2} + \cos {}^{2} (a) = 1[/tex]
[tex] \frac{3}{4} + \cos {}^{2} (a) = 1[/tex]
[tex] \cos {}^{2} (a) = \frac{1}{4} [/tex]
[tex] \cos( \alpha ) = \frac{1}{2} [/tex]
Now use the half angle identiy
[tex] \sin( \frac{ \alpha }{2} ) = \sqrt{ \frac{1 - \cos( \alpha ) }{2} } [/tex]
[tex] = \frac{ \sqrt{1 - \frac{1}{2} } }{ \sqrt{2} } [/tex]
[tex] = \frac{ \sqrt{ \frac{1}{2} } }{ \sqrt{2} } [/tex]
[tex] = \frac{ \sqrt{ \frac{1}{2} } }{ \sqrt{2} } \times \frac{ \sqrt{2} }{ \sqrt{2} } = \frac{ \sqrt{1} }{ \sqrt{4} } = \frac{1}{2} [/tex]
So the answer for our example is 1/2